Nonlinear operators. II
Nonlinear operators. II
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非线性算子。
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
C. Schwartz
中科院分区：
文献类型：
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作者：
C. Schwartz
这项工作扩展了以前关于作用于向量空间上的非线性算子的新的数学机制的发展。从通常的内积概念出发，我们发现无需引入对偶向量空间或伴随算子的概念，就可以定义厄米算子、反厄米算子和酉非线性算子。在简要研究了这些一般概念如何用于经典力学并推广量子理论的线性薛定谔方程之后，我们研究了李群和李代数的主题。该主题的许多(但不是全部)熟悉的特征被扩展到非线性算子。对于几个物理感兴趣的简单情况，找到了新的表示，并讨论了一些对基本粒子理论具有挑衅性的含义。
This work extends the previous development of new mathematical machinery for nonlinear operators acting on a vector space. Starting from the usual concept of inner product, we find that Hermitian, anti-Hermitian, and unitary nonlinear operators can be defined without bringing in the ideas of a dual vector space or adjoint operators. After looking briefly at how these general ideas might be used in classical mechanics and to extend the linear Schrodinger equation of quantum theory, the topic of Lie groups and Lie algebras is studied. Many, but not all, of the familiar features of that topic are extended to nonlinear operators. New representations are found for a few simple cases of interest to physics, and some provocative implications for elementary particle theory are discussed.